experimental investigations of rise behavior of monodispersed/polydispersed bubbly flows in...

Experimental Investigations of Rise Behavior of Monodispersed/PolydispersedBubbly Flows in Quiescent Liquids

Swapna S. Rabha and Vivek V. Buwa*

Department of Chemical Engineering, Indian Institute of Technology-Delhi, New Delhi 110 016, India

The predictive capabilities of continuum CFD models to simulate large-scale dispersed gas-liquid flowsdepend on the closures used to estimate the interphase coupling forces. The present manuscript shows thatdifferent corrections that are applied to correct the drag force for multiple bubble systems lead to differentpredictions as the gas volume fraction is increased. In the present work, experimental investigations ofmonodispersed and polydispersed bubbles of different diameters (1.2 e dB e 7.5 mm) rising in quiescentwater (0.19 e Eo e 8.72; log Mo ) -10.5) at different gas volume fractions (0.01 < RG < 0.2) are reported.The bubble rise velocity of a single isolated bubble, a bubble rising in a single chain and bubbles rising inmultiple chains were compared. The effect of bubble diameter and gas volume fraction on the fluctuations inbubble rise velocities of individual bubbles rising in multiple chains was also investigated. The rise velocitiesof monodispersed bubble swarms were found to increase with the increase in dB and RG. The number- andtime-averaged bubble rise velocity and drag coefficient for monodispersed bubble swarms were investigatedas a function of RG. The drag coefficients based on the slip velocity of the bubble swarms for RG < 0.1 werefound to decrease with increase in RG and showed good agreement with the previous literature; but for RG >0.1, the drag coefficient were found to be independent of RG. Further, the rise behavior of poly dispersedbubbles was also investigated.

1. Introduction

Several engineering processes involve dispersed gas-liquidflows, for example, in chemical, oil and gas, biochemicalindustries. Among several different contactors/reactors used forprocesses involving gas-liquid flows, bubble columns andstirred vessels are used most widely. Over the last two decades,significant research efforts were made to develop computationalfluid dynamics (CFD) models based on continuum (Euler/Euler)and discrete particle (Euler/Lagrange) approaches for detailedsimulations of unsteady dispersed gas-liquid flows in the above-mentioned gas-liquid reactors/contactors. In these approaches,the interphase momentum exchange between the two phases isaccounted through various interfacial forces, for example, drag,lift, and virtual mass forces. Several researchers1-10 couldpredict the dynamic and time-averaged characteristics of thedispersed gas-liquid flows at low gas velocities (VG < 5 cm/s)using the closure laws developed for estimation of dragcoefficient acting on a single bubble.11-18

However, for the predictions of dynamic and time-averagedproperties of dispersed gas-liquid flows at higher superficialgas velocities (VG > 10 cm/s), different empirical correctionswere applied, to the correlations used to estimate drag coefficientfor a single bubble, to account for the effect of presence of theneighboring bubbles. Rampure et al.,19 Olmos et al.,20 Kaushikand Buwa,21 Simonnet et al.22 simulated unsteady dispersedgas-liquid flows at higher superficial gas velocities (10 < VG <40 cm/s) using the drag correlation proposed by Ishii andZuber16 with a correction factor (for example,CD ) CDo(1 -RG)P). The parameter “p” was empirically tuned to higher values(3 and 4) as the gas velocity was increased. This indicates thatcontinuum models, based on closure laws that were developedfor single isolated bubbles, can predict the dynamic and time-averaged characteristics of gas-liquid flows quantitatively for

simple systems with low gas volume fractions (VG < 10 cm/s),but often fail to predict the dynamic and time- averagedcharacteristics of dispersed gas-liquid flows at higher gasvolume fractions (VG > 10 cm/s) quantitatively. One of theimportant reasons for this appear to be the lack of adequateclosure models that can account for the effect of bubble shape/size on different forces (like drag, lift and virtual mass forces)acting on bubbles and more importantly the influence ofneighboring bubbles (or volume fraction) on the magnitude ofthese interphase coupling forces.

The present work is focused on experimental investigationsof rise behavior of monodispersed and polydispersed bubblesin a (initially) quiescent liquid. The objective of the presentwork is to quantify the effect of the gas volume fraction (orbubble-bubble interactions) on the mean and the fluctuatingbubble rise velocity and finally on the drag coefficient formonodispersed and polydispersed bubbles of different sizes. Inthe following section, we bring out the present state-of-art ofthe investigations on the drag correlations developed for single/multibubble systems and of the dynamics of microscopic bubblyflows that are relevant to the present study.

2. Present State of Art

Over the years, several correlations were developed toestimate the drag coefficient (CD). Most of these correlationswere deduced either from experimental or from analyticalinvestigations of rise of a single isolated bubble and they accountfor the effects of the bubble shape and size (dB) and the slipvelocity on CD. The important correlations for CD of a singlebubble are summarized in Table 1. All these correlationsdescribe CD as a function of the bubble Reynolds number (ReB),except the correlations of Ishii and Zuber16 and Clift et al.12

which describe CD as a function of the Eotvos number (Eo).For ReB < 10, the CD predicted from the above-mentionedcorrelations decreases linearly with Re (see Figure 1). However,for ReB > 10, some correlations predict CD to be independent

* To whom all correspondence should be addressed. Tel: +9111 2659 1027. Fax: +91 11 2658 1120. E-mail: [email protected].

Ind. Eng. Chem. Res. 2010, 49, 10615–10626 10615

10.1021/ie1006454 2010 American Chemical SocietyPublished on Web 10/07/2010

Table 1. Summary of Correlations Available for Estimation of Drag Coefficient (CD) for a Single Bubble Rise

investigators correlation range of applicability

Schiller and Naumann15

CD ) { 24ReB

(1 + 0.15ReB0.687)

0.44

ReB < 1000ReB > 1000

Dalla Villle28

CD ) (0.63 + 4.8

√ReB)2

Morsi and Alexander29

CD )A

ReB+

B

ReB2 + C

0.1 < ReB < 5 × 104

Re A B C0-0.1 24 0 00.1-1 22.73 0.0903 3.69<10 29.2 -3.9 1.222<100 46.5 -116.7 0.6167<1000 96.3 2778 0.3644

Clift et al.12

CD ) 0.6221

Eo+ 0.235

1. < ReB < 10-4

0.5 < Eo < 1000

1 < log Mo < -14

Ishii and Zuber16

CD ) 2

3√Eo

0.5 < Eo < 1000

Bhaga and Weber30CD ) [(2.67)0.9 + (16/ReB)0.9]1/0.9 8.67 < Eo < 116

711 < Mo < 5.48 × 10-3

ReB < 1000

Ma and Ahmadi31

CD ) 24ReB

(1 + 0.1ReB0.75)

10 < SOReB < ∞

Lain et al.32

CD ) { 16ReB

ReB < 1.5

14.9ReB-0.78 1.5 < ReB < 80

48Re

(1 - 2.21ReB-0.5) 80 < ReB < 700

40 e Eo e 204-log Mo ) 10.5

Krishna et al.1,33 Krishna and van Baten2

CD ) 48

F1 - F2

F1gdB

1

VB2

VB ) {VB,small ) 1.53(σgF1

)VB,large ) 0.71√gdB,large(SF)(AF)

2.18 e Eo e 176.58log Mo ) -10.5

SF ) {1 fordB,large

D< 0.125

1.13exp(dB,large

D ) for 0.125 <dB,larg e

D< 0.6

0.496� DdB,large

fordB,large

D> 0.6

AF ) 2.73 + 4.505(U2 - Utrans)

dB,large ) 0.069(U2 - Utrans)0.876

Joseph34

CD ) 83+ 14.24

ReB

We f infinityReB .1

Mei et al.37

CD ) 16ReB(1 + 2

1 + 16ReB

+ 3.315

√ReB) 1 , ReB , 1

Dijkhuizen et al.38

CD ) �( 16ReB(1 + 2

1 + 16ReB

+ 3.315

√ReB))2

+ ( 4EoEo + 9.5)2 Eo.> 0.95

ReB e 1800

10616 Ind. Eng. Chem. Res., Vol. 49, No. 21, 2010

of ReB, whereas others predict a continuous decrease in CD withincrease in ReB. The correlations of CD expressed as a functionof ReB only account for bubble size (dB) and not the bubbleshape. The effect of bubble shape (accounted through Eo) onCD was studied by Ishii and Zuber,16 Clift et al.12 and Tomiyamaet al.18 and is shown in Figure 2. For small values of Eo (Eo <1) i.e. for spherical bubble regime, the CD predicted by all thesethree correlations increases marginally with the increase in Eo.However, for Eo > 1 (ellipsoidal and wobbling bubble regimes),the correlation of Clift et al.12 shows that the CD is almostindependent of Eo, whereas the correlation of Ishii and Zuber16

predicts a sharp increase in CD with increase in Eo.The correlations used to estimate CD for multibubble systems

are listed in Table 2. CD for multibubble systems was found tobe different from that of a single isolated bubble. Tomiyama etal.17,18 investigated experimentally the rise of bubbles for 0.083e ReB e 200; 0.13 e Eo e 30 and derived a correlation for CD

for bubbles rising in an uncontaminated clean liquid. Theircorrelation is valid only for systems with viscosity ratio lessthan 0.3. Ishii and Zuber16 formulated CD as a function of phasevolume fraction (RG). Recently, Rusche and Issa23 formulateda correlation to determine CD for dispersed gas-liquid flows athigher RG. A correction function f(RG) was defined in thecorrelation to account the presence of neighboring bubbles inmultiple bubble systems. Of the correlations listed in Table 2,the correlations proposed by Sankaranarayanan et al.24 was

developed by using microscopic simulations (performed usingthe lattice-Boltzmann method) of single bubble rise behaviorin a periodic box corresponding to different volume fractions.Since, in all of their simulations, Sankaranarayanan et al.24

considered the rise of a single bubble in a periodic box, theirCD correlation was applicable only to a regular array ofuniformly dispersed bubbles in a liquid and for RG < 0.2. Theeffect of RG on CD/CD0 predicted using the above-mentionedcorrelations is shown in Figure 3. The corrections proposed byIshii and Zuber,16 Tomiyama et al.,18 and Sankaranarayanan etal.24 (n ) -7) showed a decrease in CD/CD0 with increase inRG where as the corrections proposed by Rusche and Issa23 andSankaranarayanan et al.24 (n ) 4) showed an increase in CD/CD0 with increase in RG. There exists a large variation in thepredicted trends of CD/CD0 as a function of RG (see Figure 3).

Besides the studies on correlations for CD, a few experimentalstudies were reported in the literature that examined thedynamics of microscopic bubbly flows. However, most of thesestudies considered a uniform bubble size. Zenit et al.25

investigated the effects of RG (0 < RG < 0.2) for dB ) 1.35 mm(3.5% (Eo ) 0.25 and log Mo ) -10.5) on bubble risevelocities for nitrogen-water system. They used dual impedanceprobe to measure RG, mean and fluctuating bubble rise velocitiesand mean bubble size. The mean bubble velocity of the bubbleswarm was found to decrease with decrease in RG. The varianceof bubble rise velocity (VB

/ 2 ) (VB - VjB)2) was found to increasesharply with increase in RG for RG e 0.01 and rather moderatelyfor RGe 0.1. Martinez-Mercado et al.26 studied the rise behaviorof monodispersed bubbly flow (dB ) 1.2 - 1.5 mm) inair-water and air-water + glycerol system for 10 < ReB <500; 0.19 < Eo < 0.29 and -2.5 < log Mo < - 0.49 using thedual impedance probe and found the mean rise velocity ofbubbles to decrease with increase in RG. For a constant RG, theyfound that the mean bubble rise velocity (⟨VB⟩ ) VB(1 - RG)2.796)of the homogeneous dispersion to decrease with the increase inthe liquid viscosity. Initially, the fluctuations in bubble risevelocities were found to increase with increase in RG, butreached a constant value for RG > 0.02. Risso and Ellingsen27

investigated bubble rise velocity fluctuations for a homogeneousbubble swarm (dB ) 2.5 mm, Eo ) 0.21, log Mo ) -10.5) forvery small RG (0.005 < RG < 0.01) using double optical fiberprobe. Since the size of bubbles was small, the bubble risebehavior was found to be weakly influenced by the hydrody-namic interactions among the bubbles and the mean rise velocity(⟨VB⟩ ) 1.21VB) of the bubble swarm was found to be close tothat of a single bubble. It should also be noted that theseexperimental observations of mean and fluctuating bubble risevelocities were applicable to monodispersed bubbly flow withdB < 2.5 mm.

In addition to the mean bubble rise velocity and its fluctua-tions, liquid velocity fluctuations are also important to character-ize the dynamics of rise behavior of multibubble systems. Thereare only a few reports available in the literature on themeasurements of the liquid velocity fluctuations for multibubblesystems. Zenit et al.25 performed experimental investigationsof rise behavior of monodispersed bubbles using a hot filmanemometer to investigate the effect of RG on liquid velocityvariance. They observed the liquid velocity variance to increasewith increase in RG. Martinez-Mercado et al.26 found thefluctuations in the liquid velocity to increase with the increasein RG, and to become constant for RG > 0.02. Risso andEllingsen27 also investigated the liquid velocity fluctuations for

Figure 1. Comparison of CD as a function of ReB, estimated using variouscorrelations available in the literature (see Table 1).

Figure 2. Comparison of CD as a function of Eo using various correlationsavailable in the literature (see Table 1).

Ind. Eng. Chem. Res., Vol. 49, No. 21, 2010 10617

a homogeneous swarm of bubbles (dB ) 2.5 mm) for very smallvalues of RG (0.005 < RG < 0.01) using a dual tip optical fiberprobe.

From the previous literature on the investigations of micro-scopic bubbly flows, it can be noted that bubble size consideredwas very small (dB < 2.5 mm) and fall in the spherical bubbleregime of the bubble diagram of Clift et al.12 In most of theprevious work, homogeneous (monodispersed) bubbly flow wasconsidered. The rise behavior of monodispersed and polydispersed bubbles in the wobbling and spherical cap regimeswhich is more relevant to large-scale gas-liquid flows is notinvestigated. In this paper, we report the experimental investiga-tions of rise behavior of monodispersed and polydispersedbubbles for air-water system. The effect of RG (0.01 < RG <0.2) on mean bubble rise velocity and its fluctuations formonodispersed and polydispersed bubbles (1.2 e dB e 7.5 mm)

for air-water system (0.19 e Eo e 8.72 and log Mo ) -10.5)is investigated in detail. A detailed description of the experi-mental setup and benchmarking of the measurement techniqueused in the present work is provided in Section 3 and the resultsare discussed in Section 4.

3. Experimental Setup

The schematic of the experimental setup used in the presentwork is shown in Figure 4. It consists of a rectangular glasscolumn of 600 mm height with a square cross-section of 150mm width. The column was filled with demineralized water upto a height of 450 mm, and air was introduced at the bottom ofthe column through an array of stainless steel capillaries ofdifferent sizes (100 mm long, 0.26-10 mm inside diameter) togenerate bubbles of different size. Experiments were carried outwith bubble dispersions with average (sphere-equivalent) bubblediameters of 1.2 ((0.05), 2.9 ((0.1), 4.85 ((0.1), and 7.5((0.1) mm corresponding to the gas volume fraction of 0.01 <RG < 0.2. The characteristic dimensionless number for this rangeof parameters are 120 e ReB e 1821; 0.19 e Eo e 8.72; logMo ) -10.5 which correspond to the spherical, ellipsoidal, andwobbling bubble regimes. The rise behavior of single isolatedbubbles/bubble chains was recorded by using a high speedcamera (Fastec Imaging, USA) which is capable of capturing500 fps at a resolution of 1.1 megapixels. Typical experimentalimages of the bubble dispersions (dB ) 1.5 ( 0.2 mm at RG )0.03, dB ) 3.3 ( 0.1 mm at RG ) 0.09 and dB ) 4.75 ( 0.5mm at RG ) 0.16) are shown in Figure 5. The equivalent bubblediameter (dB) was calculated as dB ) 3�dmax

2 dmin, where dmax isthe major axis and dmin is the minor axis of the oblate bubble.The image processing software (Image J) was used to mea-sure the dmax and dmin of bubbles, and the average equivalentbubble diameter was calculated by taking an average over 20bubbles just after their detachment from the needles.

In the present work, the experiments were performed with a2-D array of bubbles (as shown in Figure 5) and therefore the

Table 2. Summary of Correlations Available for Estimation of Drag Coefficient (CD) for Multiple Bubble Rise

investigators correlation range of applicability

Richardson and Zaki35CD ) 0.44RG

-2.65 ReB < 1000

RG > 0.8

Ishii and Zuber16CD ) CDo(1 - RG)2 ReB ) 0.2 - 1000

Tomiyama et al.17,18

CD ) max[ 24ReB

(1 + 0.15ReB0.687),

83(1 + 17.67RG

9/7

18.67RG3/2 ) Eo

Eo + 4 ] 0.13 e Eo < 30

-10.0 < logMo < 2

0.083 e ReB< 200

Rusche and Issa23CD ) CDOf(RG);f(RG) ) exp(K1RG) + RG

K2 ReB g 1000

(K1 ) 3.64 and K2 ) 0.864)Sankaranarayanan et al.24

CD ) 43(∆F

FL)( gd3

υ2ReRReB)( 1

(1 - RG)n-1) 0.38 e Eo e 9.1

where “n’” is the Richardson-Zaki exponent -9.4 < log Mo < -4

n ) {3.3 - 1.7log( �1.3) � < 1.3

3.3 - 51log( �1.3) � g 1.3

100 < ReB < 400

RG < 0.2

� ) ReBMo0.25Eo-0.5

Figure 3. Effect of RG on CD/CDo predicted using various correlationsavailable in the literature (ReB ) 0.083 - 1000; 0.13 < Eo < 30) (seeTable 2).


area fraction of bubbles in the interrogation window wasconsidered to be a realistic estimate of the gas volume fraction.In case of 3-D array of bubbles generated by rows of needlesarranged one behind the other, it would have been appropriateto consider the actual volume fraction of bubbles. In a 2-D arrayof bubble, since the interactions from front and rear array ofbubbles (along the column depth) are absent, it is appropriateto calculate RG from the area fraction of bubbles. To measurethe area fraction, an interrogation area (AT) of different crosssections 50 × 50, 75 × 75, 100 × 100, and 125 × 125 mm2

was considered. The number of bubbles (NB) present in eachinterrogation area were counted, and the area fraction of therespective interrogation area was calculated as RG ) (πdB

2 )NB/AT. The area fraction calculations were repeated for differentareas of interrogation window until the area fractions of became

independent of AT. The vertical and horizontal components ofrise velocity of individual bubbles were measured by followingthe trajectories of the respective bubble.

Initial experiments were conducted to measure the risevelocity of single air bubbles in quiescent demineralized waterfor dB ) 1.2 ((0.05), 2.9 ((0.1), 4.85 ((0.1), and 7.5 ((0.1)mm to benchmark the measurements. A single air bubble wasreleased at the center of the column (the distance between thewall and the bubble interface was kept sufficiently large tominimize the wall effects). The rise velocities of bubbles withdB of 1.2, 2.9, 4.85, and 7.5 mm bubble were found to be 10.9,14.41, 18.76, and 23.2 cm/s, respectively and showed goodagreement with the terminal rise velocity measurements of Cliftet al.12 for contaminated water. The effect of dimensionlessliquid height (H/D) on the bubble rise velocity was studied byperforming experiments of single bubble (dB ) 3.973 ( 0.05mm) rise in a quiescent liquid (water) for three liquid heights15 cm (H/D ) 1), 30 cm (H/D ) 2), and 45 cm (H/D ) 3).The effect of liquid height on the bubble rise velocity (for dB

) 3.973 ( 0.05 mm) was found to be insignificant (results notshown here). In all further experiments, a liquid height of 45cm (H/D ) 3) was used.

4. Results and Discussion

Experiments were performed with air-water system (0.19e Eo e 8.72; log Mo ) -10.5) for different bubble size (1.2( 0.05 e dB e 7.5 ( 0.5 mm) and gas volume fraction (0.01< RG < 0.20) and the results are discussed in the followingsections. In the first part, the rise behavior of monodispersedbubbly flow is discussed which also include results on effectof neighboring bubbles (or RG) on fluctuating and mean bubblerise velocities and on CD. In the second part, the rise behaviorof polydispersed bubbly flow is discussed.

4.1. Monodispersed (Homogeneous) Bubbles Rising inQuiescent Liquid. 4.1.1. Effect of Gas Volume Fractionon Fluctuating Bubble Rise Velocities. The effects of gasvolume fraction (RG) and bubble diameter (dB) on fluctuating

Figure 4. Schematic of the experimental setup used in the present work.

Figure 5. Typical experimental images of the monodispersed bubbly flow(a) dB ) 1.5 ( 0.2 mm at RG ) 0.03, (b) dB ) 3.3 ( 0.2 mm at RG ) 0.09,(c) dB ) 4.75 ( 0.5 mm at RG ) 0.16.


bubble rise velocities of monodispersed bubbles rising inquiescent water (0.19 < Eo < 8.72; log Mo ) -10.5) werestudied by performing experiments for dB ) 1.5 ( 0.2, 3.3 (0.2, and 4.75 ( 0.5 mm at different gas volume fractions (RG

) 0.015, 0.03 for dB ) 1.5 ( 0.2 mm; RG ) 0.02, 0.06, 0.09for dB ) 3.3 ( 0.2 mm and RG ) 0.02, 0.05, 0.10, 0.16 for dB

) 4.75 ( 0.5 mm). The comparison of rise velocities of a singleisolated bubble, bubbles in a single chain and bubbles in multiple

chains for dB ) 1.5 ( 0.2, 3.3 ( 0.2 mm and 4.75 ( 0.5 mmare shown in Figures 6-8, respectively. In all these figures,the lines corresponding to the rise velocities of single isolatedbubbles indicate that measurements were repeated three times.The lines corresponding to the rise velocities of bubbles in asingle chain indicate the rise velocities of different bubbles ina single chain considered for the measurements. The linescorresponding to the rise velocities of bubbles in multiple chains

Figure 6. Comparison of rise velocities of isolated single bubbles (dB ) 1.2 ( 0.05 mm), bubbles in a single chain (dB ) 1.2 ( 0.05 mm) and bubbles inmultiple chains (dB ) 1.5 ( 0.2 mm, RG ) 0.03).

Figure 7. Comparison of rise velocities of isolated single bubbles (dB ) 2.98 ( 0.1 mm) and bubbles in a single chain (dB ) 2.98 ( 0.1 mm) and bubblesin multiple chains (dB ) 3.3 ( 0.2 mm, RG ) 0.09).


indicate the rise velocities of different bubbles in different chainsconsidered for the measurements. Though, we take only oneexperimental data set for analysis of the rise behavior of themultiple bubbles, we have also repeated the experiments on therise behavior of multiple bubbles and the range of fluctuationsin the rise velocity remains almost the same for differentexperiments (results not shown here). Therefore, one experi-mental set was considered to be adequate for the investigationsof rise behavior of multiple bubbles. The effect of gas volumefraction (0.01< RG < 0.2) on the bubble rise velocity distributionsfor dB ) 1.5 ( 0.2, 3.3 ( 0.2, and 4.75 ( 0.5 mm are shownin Figures 9-11, respectively.

The rise velocity of a single isolated bubble (dB ) 1.2 (0.05 mm) rising in quiescent water was found to be around 0.11m/s and the fluctuations in the bubble rise velocity werenegligible (see Figure 6). As the number of bubbles increase

(or RG ) 0.03), the rise velocities of individual bubbles inmultiple chains were also increased. Since the size of bubblewas small, smaller fluctuations in the rise velocities of individualbubbles were observed (see Figure 6). As shown in Figure 9,the bubble rise velocity distributions of individual bubbles risingin multiple chains (dB ) 1.5 ( 0.2 mm) at RG ) 0.03 werefound in the range of 0.24-0.42 m/s as compared to the narrowbubble rise velocity distribution observed for a single isolatedbubble.

As the bubble size was increased to dB ) 3.3 ( 0.2 mm, therise velocity of an individual bubble rising in a single chain ofbubbles was found to be higher than that of a single isolatedbubble (VB ) 0.14 m/s) which clearly shows the effect of theleading and trailing bubbles (see Figure 7). Unlike for the singleisolated bubbles, significant fluctuations were seen in the risevelocities of the bubbles rising in multiple chains (NB ) 9).The rise behavior of bubbles in multiple chains clearly indicatesthe effect of the neighboring bubbles (interaction with leadingand trailing bubbles and with the bubbles rising along the sides).The bubble rise velocity distributions of individual bubbles (dB

) 3.3 ( 0.2 mm) in multiple chains were found to be wider(VB ) 0.2-0.4 m/s for RG ) 0.02 and VB ) 0.18-0.48 m/s forRG ) 0.09) as compared to that of a single isolated bubble (dB

) 2.98 ( 0.1 mm). It should be noted that the velocitydistributions of the bubbles rising in multiple chains at RG )0.02 and 0.09 were found to shift toward high velocity regionsas shown in Figure 10 (a) and (b). The fluctuations in the risevelocities of the individual bubbles in multiple chains were alsofound to increase with the increase in RG. For dB ) 4.75 ( 0.5mm, similar results to that of dB ) 3.3 ( 0.2 mm, were observedfor increase in the magnitude of rise velocity and also in themagnitude of fluctuations in the rise velocities with increase inRG to 0.16 (see Figure 8). As compared to the narrow risevelocity distribution (0.20-0.24 m/s) observed for a singleisolated bubble (dB ) 4.85 ( 0.1 mm), rise velocity distributionsof the individual bubbles in multiple chains were found much

Figure 8. Comparison of rise velocities of isolated single bubbles (dB ) 4.85 ( 0.1 mm), bubbles in a single chain (dB ) 4.85 ( 0.1 mm) and bubbles inmultiple chains (dB ) 4.75 ( 0.5 mm, RG ) 0.16).

Figure 9. Bubble rise velocity distribution for a single isolated bubble (dB

) 1.2 ( 0.05 mm), bubble in a single chain (dB ) 1.2 ( 0.05 mm) andbubbles rising in multiple chains (dB ) 1.5 ( 0.2 mm at RG ) 0.03).


wider (0.25 - 0.35 m/s for RG ) 0.05, 0.2 - 0.52 for RG )0.10 and 0.16) as shown in Figures 11 (a)-(c), respectively.The rise velocity distribution graphs were also found to shifttoward higher velocities. This indicates the increase in interac-tions among the bubbles with increase in RG and also withincrease with dB.

4.1.2. Effect of Gas Volume Fraction on Number- AndTime-Averaged Bubble Rise Velocity. The number-averagedbubble rise velocities of the bubbles in multiple chains in ahomogeneous bubble dispersion for dB ∼ 4.75 ( 0.5 mm fordifferent values of RG are shown in Figure 12. The number-averaged bubble rise velocity ⟨VB⟩ was calculated as

The fluctuations in the rise velocity of individual bubbles werefound to decrease due to the number- averaging and the ⟨VB⟩was found to increase with increase in RG. In order to quantifythe effect of RG on ⟨VB⟩, averaging was also carried out overtime to obtain the number- and time- averaged (after discardingthe initial transients) bubble rise velocity ⟨VjB⟩ of the bubbledispersion as:

Figure 13 shows the effect of RG on number- and time-averaged bubble rise velocity (⟨VjB⟩) for dB ) 4.75 ( 0.05, 3.3( 0.2, and 1.5 ( 0.2 mm. The ⟨VjB⟩ of the homogeneousdispersion of small bubbles (dB ) 1.5 ( 0.2 mm) at RG ) 0.03was 0.34 m/s which shows good agreement with the predictedmean bubble rise velocity VB ) 0.33 m/s (Spelt and Sangani36)and experimentally measured mean bubble rise velocity VB )0.28 m/s (Zenit et al.25) for dB ) 1.35 ( 0.03 mm at RG )0.03. For large bubbles (dB ∼ 3.3 mm and 4.75 mm), the ⟨VjB⟩of the homogeneous dispersion was found to increase with theincrease in RG and remain almost constant for RG > 0.06 for dB

∼ 3.3 mm and RG > 0.10 for dB ∼ 4.75 mm (see Figure 13). Atlow gas volume fraction (RG < 0.06 for dB ∼ 3.3 mm and RG <0.10 for dB ∼ 4.75 mm), the bubbles were observed to rise ina regular array and the horizontal distance between the bubbles

Figure 10. Bubble rise velocity distribution of a single isolated bubble (dB

) 2.98 ( 0.2 mm), bubble in a single chain (dB ) 2.98 ( 0.2 mm) andbubbles in multiple chains for dB ) 3.3 ( 0.2 mm at (a) RG ) 0.02 and (b)RG ) 0.09.

⟨VB(t)⟩ ) 1NB

∑i)1

NB

VB,i(t) (1)

Figure 11. Bubble rise velocity distribution of a single isolated bubble (dB

) 4.85 ( 0.1 mm) bubble in a single chain (dB ) 4.85 ( 0.1 mm) andbubbles rising in multiple chains (dB ) 4.57 ( 0.5 mm) at (a) RG ) 0.05,(b) RG ) 0.10, and (c) RG ) 0.16.

⟨VjB⟩ ) 1T ∫ ⟨VB(t)⟩dt (2)


was large. Under such circumstance, the rise behavior ofindividual bubbles is influenced by the wake of the leadingbubble and thus the bubbles experience a higher rise velocitythan that of a single isolated bubble. But, due to smallerhorizontal distance between bubbles at high gas volume fraction(RG > 0.06 for dB ∼ 3.3 mm and RG > 0.10 for dB ∼ 4.75 mm),the lateral interactions among the bubbles in neighboring chainsincrease and dominate over the wakes effects induced by thebubbles in an individual bubble chain and diminishes the effectRG on the ⟨VjB⟩. The ⟨VjB⟩ of the homogeneous bubble dispersionwas further used to calculate the drag force (CD) and the resultsare discussed in the following section.

4.1.3. Effect of Gas Volume Fraction on Drag Coef-ficient. The drag coefficient based on slip velocity (VR ) ⟨VjB⟩- VL) of the homogeneous bubble dispersions was calculatedfrom the equation of bubble motion.

Equation 3 was simplified by neglecting lift force, virtual massforce.

However, in the absence of the instantaneous liquid velocitymeasurements for multiple bubbles rising in a (initially)quiescent liquid at different volume fractions and bubble sizes,we used the simultaneous measurements of gas and liquidvelocities reported in the previous literature.39-42 Lindken andMerzkirch39,40 measured the bubble and liquid velocities formultiple bubbles rising in a (initially) quiescent liquid (meandB ) 5.5 mm, RG ) 0.025) and reported that the mean liquidvelocity to be 5 times lower than the gas velocity (VL ) 0.2VG).Similarly, Border and Sommerfeld41 also measured the gas andliquid velocity for multiple bubbles rising in a (initially)quiescent liquid (dB ) 2 - 4 mm; RG ) 0.0075-0.0175) andfound that the mean liquid velocity to be VL ) 0.2VG Further,Sathe et al.42 measured the instantaneous bubble rise velocityand liquid velocity for dB ) 0.3-15 mm and RG ) 0.035 in a

Figure 14. CD/CD0 based on slip velocity of the bubble swarm as a functionof RG for dB ) 4.75 ( 0.5, 3.3 ( 0.2, and 1.5 ( 0.2 mm.

Figure 15. A snapshot of poly dispersed bubbles [dB1 ) 7.5 ( 0.1 mm(bubble chain at the center), dB2 and dB3 ) 4.75 ( 0.5 mm (two side chains),dB4 and dB5 ) 3.3 ( 0.2 mm (two extreme side chains)] at RG ) 0.03.

Figure 12. Number-averaged bubble rise velocity as a function time at RG

) 0.02. 0.05, 0.10, 0.16 for dB ) 4.75 ( 0.5 mm.

Figure 13. Number- and time-averaged bubble rise velocity as a functionof RG for dB ) 4.75 ( 0.5, 3.3 ( 0.2, and 1.5 ( 0.2 mm.

(FG + 0.5FL)dVbR

dt) -3

4

CD

dBFL|VbR|VbR - CLFLVbR × ∇ × VbL +

(FL - FG)gb (3)

CD ) 43(FL - FG

FL)gdB

1

VbR2

(4)


(initially) quiescent liquid. They found significantly higher liquidvelocity (VL ) 0.6VG) as compared to the liquid velocitiesreported earlier. Following the above reports, we used anapproximate VL ) 0.2VG in the calculation of CD. It should benoted that the precise effects of change in RG and dB on VL,and therefore on CD, still remains unaccounted for.

Figure 14 shows CD/CDO of the monodispersed bubbles as afunction of RG for dB ) 1.5 ( 0.2, 3.3 ( 0.2 and 4.75 ( 0.5mm. CD0 is the drag coefficient of a single isolated bubblecalculated using the measured rise velocity of that bubble. Atlow RG, CD/CD0 decreases sharply with increase in RG. Withfurther increase in RG, CD/CDO became almost constant and was0.32 for dB ) 4.75 mm, 0.19 for dB ) 3.3 mm and 0.12 for dB

) 1.5 mm.The experimental observations reported in the present work

at low gas volume fraction (RG < 0.1) agree reasonably wellwith the corrections proposed by Ishii and Zuber,16 Tomiyamaet al.18 and Sankaranarayanan et al.24 (n ) -7) which reporteda decrease in CD/CD0 with increase RG (see Figure 3). But athigher RG, our experimental observations showed CD/CDO toremain constant with further increase in RG. In most of theprevious investigations (see Table 2 and Figure 3), CD/CDO wascorrected using some function of RG to account for the effectof neighboring bubbles. It should also be noted from the presentinvestigations that the correction applied to CD/CDO to accountfor the effect of neighboring bubbles is not only a function ofRG, but is also a function of dB. Further experimental investiga-tions are in progress to study the effect of dB (importantly forlarger bubbles dB > 5 mm) and the effect of liquid propertiessuch that quantitative correction factors can be proposed.

4.2. Polydispersed (Heterogeneous) Bubbles Rising inQuiescent Liquid. In order to investigate the rise behavior ofpolydispersed bubble chains, experiments were performed withfive chains of three different bubble sizes (dB1 ) 7.5 mm, dB2

and dB3 ) 4.75 mm, dB4 and dB5 ) 3.3 mm) dispersed inquiescent water in an arrangement shown in Figure 15. The

individual bubble rise velocities in polydispersed system at RG

) 0.03 and the rise velocities of single isolated bubbles ofcorresponding dB are shown in Figure 16.

Unlike quasi-steady rise velocities observed for a singleisolated bubble with small fluctuations (see Figure 16), the risevelocity of the large bubble dB1 (∼ 7.5 mm) does not attain aquasi-steady state for the liquid height considered in the presentwork. Since the bubbles dB2 and dB3 () 4.75 mm) were releasedon each side of dB1 () 7.5 mm), the effect of the wake inducedby the large bubble dB1 on the bubbles dB2 and dB3 was morepronounced. As a result, the rise velocities of dB2 and dB3

(individual bubble rising in a chain on each side of dB1) werealso not found to reach the quasi-steady state and were almostequal to that of dB1. The bubbles dB4 and dB5 () 3.3 mm) wereless affected by dB1 and their number- and time-averaged bubblerise velocities were found to be 0.29 and 0.32 m/s respectivelywhich were close to that of a single bubble rising in a singlechain (VjB ) 0.32 shown in Figure 7). To quantify the effect ofpoly dispersity, an average CD based on slip velocity (VR )0.29 m/s) and on average <dB> () 5.16 mm) of the bubbledispersion at RG ) 0.03 was calculated and was found to bearound 0.77. Further investigations to quantify the effect of polydispersity on rise behavior of bubbles rising at different volumefractions are in progress and will be reported separately.

5. Conclusions

The rise behaviors of both monodispersed/polydispersedbubbles of different diameters (1.2 e dB e 7.5 mm) rising inquiescent water (0.19 e Eo e 8.72; log Mo ) -10.5) atdifferent gas volume fractions (0.01 < RG < 0.2) were experi-mentally investigated. The effect of neighboring bubbles on thebubble rise velocities of the individual bubbles rising in multiplechains at different gas volume fractions for different bubblediameters was investigated. Further, the CD based on slipvelocity of the bubble dispersion for different dB and RG was

Figure 16. Comparison of rise velocities of single isolated single bubbles (dB ) 2.98, 4.85, and 7.84 mm) and bubbles in multiple chains for a polydispersedsystem (dB ) 3.3, 4.75, and 7.5 mm, RG ) 0.03).


investigated. The effect of neighboring bubbles on the bubblerise velocity and on CD of the individual bubbles rising in apolydispersed system was investigated. The key conclusions ofthe present work are as follows.

For monodispersed bubbles rising in quiescent liquid, thebubble rise velocity of the individual bubbles rising in multiplechains was found to be higher than that of bubbles rising in asingle chain and single isolated bubble. The fluctuations inbubble rise velocities of individual bubbles rising in monodis-persed bubbly flows were found to increase with increase inRG and also with increase in dB. The ⟨VjB⟩ of the monodispersedbubbles was found to increase with increase in RG and remainalmost constant for RG > 0.06 for dB ∼ 3.3 mm and RG > 0.10for dB ∼ 4.75 mm. The CD based on VjR of the monodispersedbubble swarm at low RG was found to decrease with increasein RG and agreed well with the literature reports.16,18,24 But atRG > 0.1, CD was found to be independent of RG. The CD/CD0

of a monodispersed system was found to be a function of RG

and dB. Unlike monodispersed bubbly flows (for bubbles withdB < 5 mm), rise velocities of bubble in a polydispersed systemdid not attain a quasi-steady state.

Further work is in progress on systematic investigations ofrise behavior of monodispersed and polydispersed bubbly flowsto study the effect of bubble diameter (dB > 5 mm) and RG onthe CD. Such information will facilitate the development ofclosures that can account for dB and RG on the magnitude ofdrag force.

Acknowledgment

S.S.R. is grateful to the Ministry of Human ResourceDevelopment (MHRD), India for providing the researchfellowship.

Appendix

Notations

AT ) area of the interrogation window, mm2

CD ) drag coefficient, -CDo ) drag coefficient of a single isolated bubble, -CL ) lift coefficient, -D ) column depth, mmdB ) sphere equivalent bubble diameter, mmEo ) Eotvos number () g∆dB

2/σ), -g ) gravitational constant, m s-2

H ) column height, mmMo ) Morton number () g∆ FL

4/FL2σ3), -

NB ) number of bubbles, -n ) Richardson-Zaki exponent, -ReB ) Reynold number () VBFLdB/µL), -So ) density ratio, -t ) time, sVB ) bubble rise velocity, ms-1

⟨VB⟩ ) number-averaged bubble rise velocity, ms-1

⟨VjB⟩ ) number- and time-averaged bubble rise velocity, ms-1

We ) Weber number () VB2FLdB/σ), -

VL ) liquid velocity, ms-1

VR ) relative or slip velocity () VG - VL), ms-1

Greek Letters

R ) volume fraction, -µ ) molecular viscosity, Pa · sF ) density, kgm-3

σ ) surface tension, Nm-1

Subscripts and Superscripts

G ) gasL ) liquidB ) bubble

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ReceiVed for reView March 16, 2010ReVised manuscript receiVed August 9, 2010

Accepted August 25, 2010

IE1006454


experimental investigations of rise behavior of monodispersed/polydispersed bubbly flows in...

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